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Full text of "Analytical Mechanics"

WORK 171 placement and F the force on the piston then the wort done is ,JJ7 _ , dW = - F dx = - pA dx = - p dv, where p is the pressure in the fluid, A the area of the piston and dv the change in the volume of the fluid. Therefore th< total work done in compressing the fluid from a volume v to a volume t?2 is W= - I °"p<fo. (!' Jfl! When the law connecting p and v is given the work dom in compressing or expanding a fluid can be found by carry ing out the integration indicated in equation (1). During expansion, however, the displacement has the same direction as the force which causes the expansion; therefore the sign before the integral is positive. 141. Representation of the Work Done in the PV-Dia-gram.— When the volume of the expanding fluid is plotted as abscissa and the pressure as ordinate, a curve is obtained, which repre sents, graphically, the law connecting p and v. Such representation is called a PV-diagram. It is evident fror equation (1) that the area-bounded by the curve, the *;-axi! and the two vertical lines whose equations are v = DI an v = tb, represents the work done in compressing the flui from Vi to V*. 142. Isothermal Compression of a Gas. — If a gas is coir pressed without changing its temperature the compressio FIG. 96.